proof of Cassini’s identity

holds for all positive integers n≥2.
When n=2, we can substitute in the values for F1, F2
and F3 yielding the statement 2×1-12=(-1)2, which is true.
Now suppose that the theorem is true when n=m,
for some integerm≥2.
Recalling the recurrence relation for the Fibonacci numbers,
Fi+1=Fi+Fi-1, we have

Fm+2⁢Fm-Fm+12

=

(Fm+1+Fm)⁢Fm-(Fm+Fm-1)2

=

Fm+1⁢Fm+Fm2-Fm2-2⁢Fm⁢Fm-1-Fm-12

=

Fm+1⁢Fm-2⁢Fm⁢Fm-1-Fm-12

=

(Fm+Fm-1)⁢Fm-2⁢Fm⁢Fm-1-Fm-12

=

Fm2+Fm-1⁢Fm-2⁢Fm⁢Fm-1-Fm-12

=

Fm2-Fm⁢Fm-1-Fm-12

=

Fm2-(Fm+Fm-1)⁢Fm-1

=

Fm2-Fm+1⁢Fm-1

=

-(-1)m

by the induction hypothesis.
So we get Fm+2⁢Fm-Fm+12=(-1)m+1,
and the result is thus true for n=m+1.
The theorem now follows by induction.